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Dynamic stability of a base-excited thin orthotropic cylindrical shellwith top mass: Simulations and experiments$

N.J. Mallon a, R.H.B. Fey b,�, H. Nijmeijer b

a TNO Built Environment and Geosciences, Centre for Mechanical and Maritime Structures, PO Box 49, 2600 AA Delft, The Netherlandsb Eindhoven University of Technology, Department of Mechanical Engineering, PO Box 513, 5600 MB Eindhoven, The Netherlands

a r t i c l e i n f o

Article history:

Received 3 April 2009

Received in revised form

11 January 2010

Accepted 3 February 2010

Handling Editor: M.P. CartmellAvailable online 25 February 2010

a b s t r a c t

Considering both an experimental and a numerical approach, the dynamic stability of a

harmonically base-excited thin orthotropic cylindrical shell carrying a top mass is

examined. To be able to compare the experimentally obtained results with numerical

results, a semi-analytical coupled shaker-structure model is derived. Using the semi-

analytical model, it is shown that the dynamic stability analysis of the base-excited

cylindrical shell with top mass should be concentrated near a low frequency resonance,

corresponding to a mode, in which axial vibrations of the (cylindrical shell with) top

mass dominate. In this frequency region, the shell may exhibit an aperiodic beating type

of response, if some critical value of the amplitude of the harmonic base-excitation is

exceeded. This beating response is characterized by severe out-of-plane deformations.

The experimental results qualitatively confirm the numerical observations.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In the field of dynamic stability of structures, the research regarding cylindrical shells subjected to a harmonicexcitation in axial direction encompasses a large and significant part of the present literature, see [1–4] and referencescited therein. The majority of these analyses consider parametric instabilities of bare cylindrical shells, i.e. axi-asymmetrical vibration modes are excited through a Mathieu type of instability. In practise, a cylindrical shell is oftenemployed as a support structure for carrying a substantial top mass. In this case, a relatively low frequency resonance,corresponding to a vibration mode dominated by axial displacements of the (cylindrical shell with) top mass, i.e. a kind ofsuspension mode, is introduced.

In [5], it is shown that for the case of a top mass, which can only move in vertical direction, the dynamic stability limitsof the harmonically, axially excited cylindrical shell are no longer dictated by parametric instabilities in the high frequencyrange (near excitation frequencies equal to two times the eigenfrequency of an axi-asymmetrical vibration mode). Instead,the dynamic stability limits are now found in the top of a low frequency resonance far below the parametric instabilityregions, corresponding to the suspension mode introduced in the previous paragraph. To be more specific, by increasingthe amplitude of the prescribed base-acceleration, the harmonic response may become unstable in the resonance peak, anda beating type of response with severe (undesired) out-of-plane deformations of the cylindrical shell may appear instead.Furthermore and similar to the case of static buckling of axially compressed cylindrical shells, it is illustrated in [5] that the

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journal homepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

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0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jsv.2010.02.007

$ This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the technology programme of the

Ministry of Economic Affairs (STW project EWO.5792).� Corresponding author. Tel.: +31 40 247 5406; fax: +31 40 246 1418.

E-mail address: [email protected] (R.H.B. Fey).

Journal of Sound and Vibration 329 (2010) 3149–3170


critical value for the amplitude of the prescribed harmonic base-acceleration, for which the harmonic response changes tothe severe post-critical response, highly depends on the initial imperfections present in the shell.

The objective of this paper is to validate these results. Hereto, experimental results for a base-excited thin cylindrical shellwith top mass will be presented and a comparison with semi-analytical results will be made. At the experimental setup used,the base-excitation is realized by supplying a harmonic input voltage to an electrodynamic shaker system. For this casethe resulting base acceleration will not be purely harmonic, will not have a constant amplitude, but will be determined by thedynamics of the shaker system carrying the thin cylindrical shell with top mass. In addition, the bare shaker table has a lowaxial stiffness and a relatively high damping, due to the effect of back-voltage. If the cylindrical shell with top mass is fixed onthe shaker table, the shaker-shell-top mass structure exhibits two low frequency resonances (low frequency compared toresonance frequencies corresponding to axi-asymmetrical modes of the cylindrical shell). The vibration mode correspondingto the first resonance is dominated by the axial displacements of the shaker table. The vibration mode corresponding to thesecond resonance is dominated by axial displacements of the (cylindrical shell with) top mass. This mode has some couplingwith the heavily damped shaker table motion resulting in some additional damping. If one would neglect the shaker dynamicsand would consider a prescribed harmonic base-acceleration instead, as considered in [5], the first resonance would disappear,whereas the resonance frequency of the suspension mode of the top mass would shift to a lower frequency and would becomeless damped. From the discussion above, it will be clear that results obtained by voltage excitation can only be compared in aqualitative sense with results obtained by a prescribed harmonic base-acceleration, as considered in [5]. Therefore, to enablequantitative comparison between experimental results and semi-analytical results in the current paper, the semi-analyticalmodel of the base-excited cylindrical shell with top mass will be coupled to a model of the electrodynamic shaker.Furthermore, the cylindrical shell made from Poly Ethylene Terephthalate appeared to behave orthotropic in the elasticdomain. Therefore, in the current paper orthotropic material behaviour will be introduced in the semi-analytical modellingapproach developed in [5] for isotropic cylindrical shells.

Experimental results considering a base-excited cylindrical shell with a free top mass, i.e. the top mass is onlysupported by the cylindrical shell, are presented in [6]. Near the resonance of the first axi-symmetric vibration mode, avery severe unstationary response is found. In this region, the base-acceleration due to the shaker could not be controlledto remain purely harmonic. Furthermore, the obtained results could not be explained using numerical simulations.Consequently, a combined numerical and experimental analysis of a base-excited cylindrical shell carrying a top mass, aswill be presented in this paper, has not been previously presented.

The outline for this paper is as follows. In the next section, the experimental setup of the cylindrical shell with top masswill be introduced and the material properties of the cylindrical shell will be determined. In Section 3, the coupled shaker-structure model will be discussed. In Section 4, modal analysis and static buckling analysis will be performed and resultswill be compared with finite element analysis (FEA) results. The theoretical modal analysis results will be compared withexperimental results. Dynamic stability of the cylindrical shell with top mass excited by the shaker will be studiednumerically in Section 5 and experimentally in Section 6. Finally, in Section 7 conclusions will be presented.

2. Experimental setup

The seamless cylindrical shell, which is used for the experiments, is cut out from an unused beverage bottle made ofPoly Ethylene Terephthalate (PET), see Fig. 1. The obtained cylindrical shell has radius R (measured at the neutral plane)and (average) thickness h. The shell thickness varies in axial direction by approximately 2 percent. The PET bottle isproduced by using a blow moulding technique. During this process, the material is stretched in the axial direction of thebottle and subsequently in the circumferential direction of the bottle. Since this biaxial stretching is performed withdifferent stretch ratios in the two (perpendicular) directions, a directional dependency of the elasticity properties isintroduced. The elasticity properties of a biaxially stretched, thin PET film may fairly well be approximated using anorthotropic symmetric material description, with principle directions aligned with the two stretch directions [7]. Thisapproximation is also used here to model the elasticity properties of the PET cylindrical shell. More specifically, the shell isassumed to be made of orthotropic material with principle directions e1 and e2 coinciding with, respectively, the axialcoordinate x and the circumferential coordinate y (i.e. the two stretch directions), see Fig. 1. The orthotropic shell materialmodel is described by four parameters, i.e. Young’s moduli in x and y direction (Ex and Ey), the shear modulus Gxy, and onePoisson’s ratio (either nx or ny, since nyEx ¼ nxEy). The procedure followed to identify these parameters is discussed in detailin [8]. The resulting material and geometrical parameter values are listed in Table 1.

Since the in-plane boundary conditions have a significant influence on the thin shell behaviour [9–11], special effort istaken to obtain a rigid clamping of the cylindrical edges. An exploded view of the construction used to clamp edges of thecylindrical shell is depicted in Fig. 2. The cylindrical shell edge is fixed between an inner ring, which fits exactly to the innerdiameter of the shell, and an open ring with conical outer shape. The outer ring, which has a conical inner shape, is screwedover the open ring on the inner ring. In this manner, the open ring is compressed radially on the cylindrical shell surface,resulting in a stiff circle line contact between the clamping rings and the thin shell. Air outlets at the bottom clampingstructure avoid entrapment of air in the shell.

The experimental setup is depicted in Fig. 3. The thin cylindrical shell, which is fixed between clamping rings both at itstop and at its bottom, is mounted between two linear sledges with very low friction in axial direction, see Fig. 3. At the top

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N.J. Mallon et al. / Journal of Sound and Vibration 329 (2010) 3149–31703150


side, the linear sledge is based on air bearings. At the bottom side, the linear sledge is realized by an elastic mechanismbased on elastic leaf springs. The purpose of these support mechanisms is to minimize transversal motions and rotations ofthe cylindrical shell edges. The upper linear sledge and upper clamping structure actually act as the rigid top mass (total

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R

e1

e2

x

�

Fig. 1. Cylindrical shell from PET beverage bottle.

Table 1Material and geometrical properties of the orthotropic cylindrical shell.

Ex 3.85 (GPa) Ey 6.00 (GPa) Gxy 1.94 (GPa)

nx 0.25 (–) ny 0.39 (–) r 1350 (kg/m3)

L 85 (mm) R 44 (mm) h 0.23 (mm) (R/h=191)

outer ring

cylindrical shell

open ring

inner ring

Fig. 2. Exploded view of one clamping ring for cylindrical shell.

N.J. Mallon et al. / Journal of Sound and Vibration 329 (2010) 3149–3170 3151


mass mt=4.7 kg). The moving mass of the lower linear sledge, including the mass of the bottom clamping structure and themass of the shaker armature, equals mb=4.1 kg.

The axial base-excitation is realized using an electrodynamic shaker system (type LDS PA1000L/LDS V455), see Fig. 3. Aperiodic excitation is introduced by supplying a harmonically varying input voltage

E0ðtÞ ¼ vdsinð2pftÞ ½V�; (1)

to the power amplifier, which output voltage is supplied to the shaker. The amplifier works in a voltage mode of operation,i.e. the output voltage of the amplifier is kept proportional to its input voltage. No active feedback is used to control theacceleration of the shaker armature €Ub. Consequently, as stated before, the resulting acceleration of the shaker (and thusthe effective axial force on the cylindrical shell with top mass) will not be proportional to Eq. (1), but will be determined bythe dynamics of the electro-mechanical shaker system carrying the cylindrical shell with top mass.

For dynamic response measurements, two sensors are used. Firstly, the relative axial displacement of the top mass Ut(t),see Fig. 3, is measured using a Shaevitz 100 MHR linear variable differential transformer (LVDT). Furthermore, a laservibrometer (Ono Sokki LV 1500) is used to measure the transversal velocity _w at one point of the cylindrical shell. Thesignal of the laser vibrometer is numerically integrated to obtain measurements in terms of transversal displacements w.To avoid drift during the numerical integration, the measurement signal is filtered using a high-pass filter with a cut-offfrequency of f ¼ 1:6 Hz. The data-acquisition and input signal generation is performed using a laptop equipped withMatlab/Simulink in combination with a TUeDACS AQI [12]. The sample frequency used is 4 kHz.

3. Modelling approach

In this section, the equations of motion are derived for the thin orthotropic cylindrical shell carrying a rigid top mass mt.Since the cylindrical shell is excited at its base in axial direction by an electrodynamic shaker, an electro-mechanical model

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Ub (t)

Ut (t)

a

b

c

d

e

f

Power amplifierE0 (t)

Fig. 3. Picture and schematic overview of the experimental setup ((a) top linear sledge based on air bearings, (b) linear variable differential transformer

(LVDT), (c) cylindrical shell, (d) laser vibrometer, (e) bottom elastic support mechanism, (f) electrodynamic shaker).



of the shaker will be included in the semi-analytical model, in order to be able to compare the experimental results withthe semi-analytical results. The modelling of the orthotropic cylindrical shell with top mass is based on the sameassumptions adopted in [5] for the modelling of an isotropic cylindrical shell with top mass.

3.1. Cylindrical shell model

The dimensions of the cylindrical shell are defined by the radius of the neutral plane R, thickness h, and length L.Considering the cylindrical coordinate system ½r¼ R; x; y� defined in Fig. 1, the axial in-plane displacement field is denotedby uðt; x; yÞ, the circumferential in-plane displacement field by vðt; x;yÞ, the radial out-of-plane displacement field bywðt; x; yÞ, and the radial imperfection shape by w0ðx; yÞ. For readability, the notations for the displacement fields and radialimperfection shape will be abbreviated to u, v, w, and w0, respectively. The axial coordinate x and axial displacement field u

are measured relative to the base-motion Ub (t).According to Donnell’s assumptions, the nonlinear strain–displacement relations read [13,14]

ex ¼ u;xþ12w;2x þw;xw0;x; kx ¼�w;xx;

ey ¼1

Rðv;yþwÞþ

1

2R2w;2y þ

1

R2w;yw0;y; ky ¼�

1

R2w;yy;

gxy ¼1

Ru;yþv;xþ

1

Rðw;xw;yþw;xw0;yþw0;xw;yÞ; kxy ¼�

1

Rw;xy; (2)

where ,x means q=qx and ;y means q=qy. Note that in Eq. (2), the radial displacement field w and the radial imperfectionshape w0 are measured positively inwards.

Considering orthotropic material properties with principle axes, which coincide with the cylindrical coordinate axes,the stress resultants and stress couples per unit length are defined by [15,16]

Nx

Ny

Nxy

264

375¼ h

C11 C12 0

C21 C22 0

0 0 C33

264

375

ex

eygxy

264

375;

Mx

My

Mxy

264

375¼ h3

12

C11 C12 0

C21 C22 0

0 0 C33

264

375

kx

ky

kxy

264

375; (3)

where

C11 ¼Ex

1�nxny; C22 ¼

Ey

1�nxny; C33 ¼ Gxy; C12 ¼ C21 ¼ ny

Ex

1�nxny: (4)

The following boundary conditions for the cylindrical shell with rigid end-disks are considered (‘–’ means notprescribed)

(5)

Note that the base-motion does not appear in the boundary conditions, since u and x are measured relativeto UbðtÞ. Indeed, for a thin cylindrical shell mounted between two rigid end-disks, the clamping condition (in practice)is probably closer to the condition w,x = 0 than to the condition Mx = 0. Nevertheless, the clamping condition is assumedto be Mx = 0, since this allows for a simple expansion of w. This assumption is supported by the fact, that previousstudies show, that the rotational boundary condition (Mx = 0 vs. w,x = 0) has only a mild influence on the eigenfrequencies[9,10], parametric instabilities [17], and nonlinear vibrations [10,11]. The boundary condition in terms of themembrane force Nx at x=L, due to the inertia force of the top mass, is not included. This force will be included via thekinetic energy.

The strain energy of the structure, corresponding to Donnell’s assumptions, reads as follows:

Us ¼1

2

Z 2p

0

Z L

0ðNxexþNyeyþNxygxyÞdx R dyþ

1

2

Z 2p

0

Z L

0ðMxkxþMykyþMxykxyÞdx R dy: (6)

The case mt bmshell, where mt is the top mass and mshell is the mass of the shell, is considered. Therefore, the contribution ofthe mass of the shell is neglected in the gravitational potential energy of the structure

Ug ¼mtgðUbðtÞþuðt; L; yÞÞ; (7)

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where g is the acceleration due to gravity. Furthermore, the influence of in-plane inertia of the shell is also neglected in thekinetic energy

T cyl ¼1

2rh

Z 2p

0

Z L

0

_w2 dxR dyþ1

2mt _u

2t ; (8)

where _ut ¼_UbðtÞþ _uðt; L; yÞ.

Based on Eqs. (6)–(8), the nonlinear equilibrium equations become

RNx;xþNxy ¼ 0; (9)

RNxyþNy;y ¼ 0; (10)

Mx;xxþ2

RMxy;xyþ

1

R2My;yyþ

1

RNyþNxðw;xxþw0;xxÞþ

2

RNxyðw;xyþw0;xyÞþ

1

R2Nyðw;yyþw0;yyÞ ¼ rh €w: (11)

Since the effects of in-plane inertia are neglected, these equations constitute a set of two static in-plane equilibriumequations (Eqs. (9)–(10)) and one dynamic out-of-plane equilibrium equation (Eq. (11)).

The out-of-plane displacement field is expanded as

wðt; x; yÞ ¼XN

i ¼ 1

XMj ¼ 0

½Q sijðtÞsinðjnyÞþQ c

ijðtÞcosðjnyÞ�sinðlixÞ; (12)

where li ¼ ip=L, i is the number of axial half-waves, n is the number of circumferential waves, and Q s;cij ðtÞ are N(2M+1)

generalized degrees of freedom (DOFs). Note that Eq. (12) exactly satisfies the boundary conditions for w, see Eq. (5). The N

DOFs Q ci0ðtÞ correspond to axi-symmetrical radial displacements and the 2NM DOFs Q s;c

ij ðtÞ ðja0Þ to axi-asymmetricaldisplacement fields. The presence of pairs of companion modes (related to DOFs Qij

s and DOFs Qijc for ja0) with the same

shape, but with a different angular orientation, is due to axi-symmetry of the (perfect) shell [2–4,18].The following (axi-asymmetrical) expansion of the radial imperfection w0 is considered

w0ðx; yÞ ¼ hXNe

i ¼ 1

eisinðnyÞsinipx

L

� �; (13)

where NerN and ei are dimensionless imperfection amplitudes.Using the above expressions for w and w0, the in-plane equilibrium equations (Eqs. (9)–(10)) now consist of a set of

linear coupled inhomogeneous partial differential equations (PDEs) in terms of u and v. In order to perform a reductionof the three independent displacement fields to one independent displacement field w, these two PDEs are solvedsymbolically. The solution procedure for this purpose is outlined in [8] and results in expressions for u and v, which arefunctions of w and w0. These expressions exactly satisfy the in-plane equilibrium equations (Eqs. (9)–(10)) and the in-planeboundary conditions (Eq. (5)). During this step, an extra DOF UtðtÞ ¼ uðt; L; yÞ is introduced, which corresponds to the axialdisplacement of the top mass.

3.2. Shaker model

The linear model of the electrodynamic shaker is depicted in Fig. 4, where the electrical part and the mechanical partare presented separately. The electrical part of the model (Fig. 4(a)) is described by a power amplifier, current I(t), coilresistance R, coil inductance L, current-to-force constant kc , and back voltage EbackðtÞ ¼ �kc

_Ub. The power amplifier worksin a voltage-mode of operation. More specifically, in the frequency domain

EðjoÞ ¼ GampðjoÞE0ðjoÞ; (14)

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Power amplifier

E0 (t)E (t)

Ub (t)

Gamp

R L

Eback (t)kb

cb

mb

I (t) Femf (t)

Fig. 4. Model of the electrodynamic shaker (electrical part (a), mechanical part (b)).



where GampðjoÞ (with j2 = �1) is the frequency dependent amplifier gain, E the amplifier output voltage, and E0 theamplifier input voltage.

Mass mb of the mechanical part of the model (Fig. 4(b)) corresponds to the moving mass of the lower linear sledge,including the mass of the bottom clamping and the mass of the shaker armature, see Fig. 3. This mass, which is supportedby an elastic suspension with stiffness kb, is excited by the electromagnetic force Femf ðtÞ ¼ kcIðtÞ. The vertical displacementof the mass is described by DOF Ub. Energy dissipation in the mechanical part of the shaker is modelled by a linear viscousdamping force with viscous damping constant cb. It should be noted that the shaker base is assumed to be rigidlyconnected to the fixed world. If this assumption would not be valid, the mechanical part of the shaker model would need tobe extended to a multi-DOF model, resulting in more mechanical parameters to be identified, see for example [19].

The dynamics of the shaker are now described by the following two coupled ordinary differential equations [20]

L_IþRIþkc_Ub ¼ EðtÞ; mb

€Ubþcb_UbþkbUb ¼ kcIþFstr; (15)

where E is related to E0 by Eq. (14), and Fstr is the force exerted to the shaker mass by the cylindrical shell with top mass.Force Fstr in general depends on Q s;c

ij ðtÞ, UtðtÞ, and first and second time derivatives of these DOFs. During the identificationprocedure of the unknown parameters of the excitation mechanism, in which the bare shaker was used, i.e. Fstr=0 N, afrequency dependency of the amplifier gain defined by

GampðjoÞ ¼ Pampðbamp � joþ1Þ (16)

is adopted to obtain a good fit of the combined shaker-amplifier dynamics for the frequency range of interest (0–300 Hz).The parameters of the combined shaker-amplifier model are identified using frequency domain techniques, see [8] formore details. The identified parameter values for the shaker model and the amplifier model are listed in Table 2. Note thatusing Eq. (16), the time domain version of Eq. (14) becomes EðtÞ ¼ Pampðbamp

_E0ðtÞþE0ðtÞÞ, where E0 is the known function oftime given by Eq. (1). This time domain version will be used in the next subsection.

3.3. The coupled shaker-structure model

The coupled shaker-structure model will be derived by following a charge-displacement formulation of Lagrange’sequations [21]. In this formulation, energy and work expressions of the coupled structure are formulated in terms ofmechanical DOFs and, in this case, a single additional charge coordinate q, whose time derivative constitutes the currentthrough the electrical part of the shaker model, i.e. _q ¼ I.

The total set of N(2M+1)+3 DOFs of the coupled model is collected in the column

Q � ¼ ½Q c10; . . . ;Q

sNM;Q

cNM;Ut;Ub; q�

T; (17)

where Qijs, c and Ut are the generalized DOFs of the structure and Ub is the axial motion of the shaker (see Fig. 4).

Linear viscous damping is included in the cylindrical shell structure via the following Rayleigh dissipation function:

Rcyl ¼1

2

XN

i ¼ 1

XMj ¼ 0

cijð_Q

c

ijÞ2þ

1

2

XN

i ¼ 1

XMk ¼ 1

cikð_Q

s

ikÞ2þ

1

2ct_U

2

t ; (18)

where cij, cik, and ct are positive constants. Furthermore, in the model of the cylindrical shell structure, the axial motions aredefined with respect to an arbitrary base motion Ub, see Eq. (8). For the coupled shaker-structure system, the energy andwork expressions and the Rayleigh dissipation function now become as follows:

M¼ 12L _q2þkc _qUb;

T ¼ T cylþ12mb

_U2

b;

V ¼ UsþUgþ12kbU2

b;

R¼Rcylþ12 cb

_U2

bþ12RI2;

dWnc ¼ EðtÞdq; (19)

where M is the magnetic energy of the moving coil of the shaker, and dWnc is the virtual work of E(t) [21], which is theoutput voltage of the amplifier, see the previous subsection. Defining the Lagrangian L of the complete system by

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Table 2Parameters of shaker model and amplifier model.

cb 278 (kg/s) L 2:6� 10�3 (H) Pamp �88.3 (–)

mb 3.0 (kg) kc 11.5 (N/A) bamp 1:4� 10�3 (s�1)

kb 5:28� 104 (N/m) R 0.9 ðOÞ



L¼ T þM�V, the final coupled set of equations of motion can be determined by

d

dtL; _Q ��L;Q � þR; _Q � ¼ bEðtÞ; (20)

where b = [0,y,0,1]T is an N(2M+1)+3 dimensional column. Among others, this will lead to an explicit expression for Fstr,the force exerted to the shaker mass by the cylindrical shell with top mass, which was introduced in Eq. (15).

3.4. Selection of the number of DOFs

For the sake of brevity, only one specific expansion of w, see Eq. (12), will be considered. The specific expansion isobtained by deleting, i.e. setting to zero, a number of DOFs, corresponding to the expansion of w with N=10 and M=2. Thedeleted DOFs are DOFs Qij

s, c with i=2,4,y,10, which correspond to axial-asymmetrical radial displacements, and DOFs Qi2s

and Qi1c , which are companion modes of the kept (both axi-asymmetric and axial-symmetric) modes. More specifically, for

the analyses described in this paper, the out-of-plane displacement field is expanded as follows:

wðt; x; yÞ ¼X

i ¼ 1;3;...;9

½Q ci0ðtÞþQ s

i1ðtÞsinðnyÞþQ ci2ðtÞcosð2nyÞ�sinðlixÞ: (21)

Note that for the modes with 2n waves in circumferential direction, the modes with DOFs Qi2c are selected, since these

modes are directly coupled (via a quadratic nonlinearity) to the modes with DOFs Qi1s . The modes corresponding to

(deleted) DOFs Qi2s would only posses a parametric coupling with the modes corresponding to DOFs Qi1

s .The 15-DOF expansion of w given by Eq. (21) appears to be sufficient for accurately predicting the onset to the severe

beating response, as was also found for the case of prescribed base-acceleration, see [5]. DOFs Qi0c correspond to modes

being both axi-symmetric and axial-symmetric. The mode depicted in Fig. 5(a) is a linear combination of modes of this typeand is therefore also both axi-symmetric and axial-symmetric itself. DOFs Qi1

s and Qi2c correspond to modes being both

axial-symmetric and axi-asymmetric, where DOFs Qi2c have a double number of circumferential waves compared to DOFs

Qi1s . Fig. 5(b) shows an example of a mode consisting of a linear combination of modes being both axial-symmetric and axi-

asymmetric. As stated before, axial-asymmetrical modes are not included in Eq. (21). An example of such a mode, builtfrom a linear combination of modes of this type, is visible in Fig. 5(c). As mentioned at the end of Section 3.1, during thestep where the expressions for in-plane fields are solved, an extra DOF Ut is introduced, which corresponds to the relativeaxial displacement of the top mass. After coupling to the shaker model, which is described by the two DOFs Ub and q, theresulting semi-analytical model now in total possesses 15+1+2=18 DOFs. In this model, only single mode imperfectionshapes, i.e. only one eia0, will be considered.

4. Modal analysis and static buckling analysis

To obtain insight in the eigenfrequencies and damping ratios of the shaker-structure system, modal analyses areperformed. Experimental results will be compared with semi-analytical results (using the 18-DOF model with w0=0 m)and for some cases also with FEA results, obtained via the finite element (FE) package MSC.Marc. The used FE model of thecomplete cylindrical shell consists of 30 000 four-node thin shell elements based on Kirchhoff theory. In MSC.Marc theseelements are referred to by element type 139 [22]. The nonlinear kinematic relations used in the FE model are valid forlarge displacements and moderate rotations. Both in the semi-analytical analysis and in the FEA, the effect of the preloaddue to the weight of the top mass is taken into account. In order to carry out a modal analysis, both the semi-analyticalmodel and FE model are linearized around the static equilibrium solution.

First, the eigenfrequencies corresponding to the lowest two axi-symmetric modes are determined both experimentallyand by using the semi-analytical model. No FEA results are available for these two modes, since the electrical part of theshaker could not be modelled in the FE package. For the experimental approach, frequency response functions (FRFs) are

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(axi-sym. and axial-sym.) (axi-asym. and axial-sym.) (axi-asym. and axial-asym.)

Fig. 5. Three types of vibration modes of cylindrical shell (top mass not shown).



measured by exciting the system with a randomly varying input voltage E0, while measuring the relative axialdisplacement of the top mass Ut. In order to realize a linear response, the excitation level is kept low. The resulting(averaged) FRF shows a heavily damped ðx1 ¼ 0:30Þ resonance at f ¼ f1 � 12 Hz and a moderately damped ðx2 ¼ 0:05Þresonance at f ¼ f2 � 182 Hz, see Fig. 6. The first resonance corresponds to an axial suspension mode of the shaker mass, i.e.a mode dominated by Ub. The second resonance corresponds to an axial suspension mode dominated by Ut, see Fig. 5(a).Based on the experimentally determined damping ratios, the damping parameter of the semi-analytical model ct (cij arekept zero, see Eq. (18)) is tuned to fit these ratios. In Fig. 6, also the FRF obtained using the 18-DOF semi-analytical model isdepicted. As can be noted, up to f=300 Hz the semi-analytical results are in very good agreement with the experimentalresults. The measured FRF shows small resonances near f=350 Hz, which are not present in the semi-analytical results. Aswill be shown next, these resonances are not due to axi-asymmetric modes of the cylindrical shell, since these occur above850 Hz. In addition, these high-frequency modes only posses weak coupling with the axi-symmetric modes at 12 and182 Hz, due to imperfections in the shell. Only for very large imperfections (e.g. amplitude 20 times the shell thickness),some very minor influence can be noted in the FRF in the frequency region of the shell eigenfrequencies (i.e. near 1000 Hz).Therefore, the resonances near f=350 Hz are likely due to the finite stiffness of the shaker support structure.

The eigenfrequencies and damping ratios of the lowest axi-asymmetric vibrational eigenmodes of the cylindrical shell,which are also axial-symmetric (see Fig. 5(b)), are experimentally determined for several values of n by exciting the topmass using an impulse hammer. The resulting out-of-plane velocity is measured at height x=L/2, using the laservibrometer at 40 points, which are distributed equidistantly along the circumference of the shell. Based on these FRFs,the eigenfrequencies with corresponding eigenmodes and damping ratios are determined, see [23] for more details.The experimental results are shown in the 4th and 5th column of Table 3. For n=7 no experimental results are included,since for this value of n no mode could be identified. In Table 3, also undamped eigenfrequencies of the perfect cylindricalshell structure, determined using the semi-analytical approach (using the 18-DOF model) and using FEA, are shown in,respectively, the 2nd and 3rd column. As can be noted, the semi-analytical results are in good correspondence with the FEAresults. The maximum difference is 2 percent for n=4. The semi-analytical results show an average difference of 8 percentwith the experimental results. Differences may be due to various reasons, e.g. due to imperfections, which are notaccounted for in the semi-analytical and FE model, due to inaccuracies in the identified material properties and/or due tomeasurement inaccuracies.

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10−5

10−4

10−3

0 100 200 300 400 500−10

−5

0

0 100 200 300 400 5000

0.5

1

Mag

. [m

/V]

Phas

e [r

ad]

Coh

eren

ce [

-]

Model

Experiment

f [Hz]

Fig. 6. FRF from input voltage E0(t) to relative top mass displacement Ut(t).



Next, static buckling of the cylindrical shell is considered. In [24], an analytical expression is derived for the bucklingload of orthotropic cylindrical shells with simply supported boundary conditions and being subjected to axial edge loads. IfEq. (29) from [24] is applied, the lowest buckling load for the cylindrical shell under consideration is estimated to bePc=951 N. The corresponding buckling mode is characterized by n=6 and i=1. Recall that i is the number of half-waves inaxial direction, see Eq. (12). However, due to the fact that the cylindrical shell under consideration is mounted betweenstiff end-rings, its real boundary conditions at both ends of the cylindrical shell are prescribed axial displacements ratherthan axial edge loads. This difference in the in-plane boundary condition may slightly affect the buckling load and thecorresponding buckling mode of moderately long cylindrical shells, to which class the cylindrical shell under considerationbelongs [25]. The static buckling of the orthotropic (perfect) shell is also determined using a linearized buckling eigenvalueanalysis based on the 18-DOF semi-analytical model and, as a numerical reference, by using FEA based on the same modelas used for the modal analysis. Several values of the circumferential wavenumber n are considered. The 18-DOF semi-analytical model predicts the lowest buckling load of the perfect cylindrical shell to be Pc=990 N and the correspondingbuckling mode is dominated by n=12 and i=7. The FEA predicts Pc=975 N and a corresponding buckling mode dominatedby n=1 and i=9. Note that the buckling mode found by the FEA cannot be predicted accurately by the semi-analyticalmodel, since the adopted shell theory of Donnell is only valid for nZ4. As is well-known for thin cylindrical shells [25], justabove the first critical load, many additional closely spaced buckling loads are obtained. This is confirmed once again by theresults obtained above. The buckling loads obtained using the semi-analytical approach and the FEA differ only slightly(2 percent). Based on the buckling load predicted by the FEA, the weight of the top mass, as considered during theexperiments, equals 5 percent of the static buckling load, i.e.mt � g=Pc ¼ 0:05 [–].

5. Numerical steady-state analysis

In this section, a numerical steady-state analysis will be performed for the total system consisting of the shaker, theamplifier, and the cylindrical shell with top mass. The objective of this dynamic analysis is to determine, for whichcombinations of excitation frequency f and amplitude vd of the input voltage, see Eq. (1)), instabilities occur, and how theseresults depend on possible geometric imperfections in the cylindrical shell. The amount of damping taken into accountduring the simulations is based on the experimental modal analysis results as discussed in the previous section, i.e. thedamping ratios of the lowest two (axi-symmetric) modes are set to x1 ¼ 0:3 and x2 ¼ 0:05 (related to ct and cb). To all othervibration modes a damping ratio x¼ 0:01 is assigned, which is realized via parameters cij. This value corresponds, more orless, to the average of the experimentally determined damping ratios of the lowest axi-asymmetric modes, see Table 3.

Numerical continuation of periodic solutions [26] in combination with Floquet theory will be used to determine theparameter region(s), where the stability of the harmonic response is lost. In the frequency regions, where the harmonicresponse is no longer stable, the response is further examined using standard numerical integration and a numericalimplementation of a stepped frequency sweep procedure. Detailed results are presented for a circumferential wavenumbern=9, while considering three (single mode) imperfection shapes, i.e. e1=0.5, e3=0.5, or e5=0.5. The influence of the selectedvalue for n will be addressed at the end of this section. The responses of the cylindrical shell are characterized by thefollowing dimensionless measures:

uL ¼ uðt; L; yÞ=L; (22)

Um ¼maxT

uL�minT

uLZ0; (23)

wL=2 ¼wðt; L=2;p=ð2nÞÞ=h; (24)

where T=1/f. Note that uL can be directly computed from the DOF Ut, which was introduced in the model at the end ofSection 3.1.

Using numerical continuation of periodic solutions with the excitation frequency f as continuation parameter, it isfound that for sufficiently large excitation amplitude vd, the harmonic response first loses stability in the top of the

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Table 3

Eigenfrequencies of lowest axi-asymmetric, axial-symmetric modes; both the 18-DOF and the FE model consider the undamped case for w0=0 m; xn1

denotes the experimentally estimated damping ratio.

n 18-DOF (Hz) FEA (Hz) Experiment (Hz) xn1 (–)

4 1415 1387 1293 0.005

5 1070 1063 1030 0.007

6 896 896 886 0.010

7 858 858 – –

8 929 926 997 0.020

9 1076 1070 1165 0.010

10 1276 1268 1496 0.007

11 1514 1506 1683 0.020



resonance peak near f2=182 Hz. Recall, that the resonance near f= f2 corresponds to a vibration mode, which is dominatedby axial displacements of the (cylindrical shell with) top mass, i.e. by Ut. Similar to the case of prescribed base-acceleration,discussed in [5], responses with large out-of-plane deflections appear, when the stability of the harmonic solution is lost.The critical value of vd, for which the harmonic solution loses stability for f= f2, is called vd

c .For the perfect cylindrical shell (w0=0 m), the harmonic solution loses stability in the peak of the resonance near f= f2 for

vd = vdc = 0.298 V. By introducing an imperfection of the form e1=0.5, this critical value decreases 15 percent to vd

c =0.253 V. For this case, the frequency–amplitude plots near f= f2, for vd just below and just above vd

c , are depicted in Fig. 7.As can be noted, for vd4vc

d, a very complicated branch of unstable periodic solutions appears in the resonance peaknear f= f2, which is initiated by two cyclic fold bifurcations. The response in this region of instability is further examinedusing standard numerical integration of the equations of motion, see Fig. 8. As can be noted, a beating response appearsin the region of instability. Time intervals with small out-of-plane displacements are alternated with time intervals,where the out-of-plane response wL/2 of the shell is very severe. In the latter time intervals, very high frequencies can beobserved in the transversal acceleration signal €wðL=2Þ. In very short time periods associated with these high frequencies,transversal accelerations of the shell up to 20 000g occur. In these very short time periods, these accelerations lead totransversal displacements in the order of the plate thickness, giving rise to nonlinear kinematic coupling between in-planemotion and out-of-plane motion. For these very short time periods, the previous assumption, that the in-plane inertia ofthe cylindrical shell can be neglected, may become questionable. This may lead to an overestimation of the transversalacceleration. Transversal peak accelerations of the shell surface of 2100g are measured in experiments described in [6] for amore or less comparable structure and situation; in [6], the top mass is free and five times lighter. The power spectraldensity (PSD) of the beating response is broad-banded (see bottom plot of Fig. 8), suggesting that this response has achaotic nature.

A possible explaination for the occurring instability phenomenon may be the nonlinear interaction between the axialmode near 182 Hz, see Fig. 5(a), and the shell type of modes present in the high frequency range, i.e. above approximately850 Hz, see Table 3. Note that the modal density of the shell modes is quite high. The occurring interactions may be causedby direct nonlinear couplings and/or parametric couplings present in the model. As will be shown, the instabilityphenomenon appears to be robust. It also occurs for other imperfections ei and other circumferential wavenumbers n.

The response of the total system will be examined experimentally using a stepped sine frequency sweep procedure.The obtained frequency–amplitude plots from these frequency sweep experiments cannot fully be compared with thefrequency–amplitude plots computed using the continuation approach, since the latter ones do not include aperiodicresponses (as obtained numerically for vd4vc

d). Therefore, the numerical response is also examined using animplementation of the stepped sine procedure, which is based on numerical integration of the equations of motion.During the stepped sine frequency sweep, the excitation frequency is incrementally increased (in case of sweep-up) ordecreased (in case of sweep-down) using a step size Df ¼ 0:5 Hz. For each discrete value of f, the signals are saved duringNe=150 excitation periods. The data during the first Nt=50 periods are not used, in order to minimize transient effects.

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4

4.5

5

5.5

6

6.5

7x 10−3

vd = 0.25

vd = 0.26

f [Hz]

f2

Um

[-]

stableunstable

Fig. 7. Frequency–amplitude plot of the imperfect shell (e1=0.5).



Subsequently, the following averaged response measures are determined:

~Um ¼1

Nm

XNm�1

k ¼ 0

maxTm

uLðtiþkTmÞ�minTm

uLðtiþkTmÞ

!; (25)

~W m ¼1

Nm

XNm�1

k ¼ 0

maxTm

wL=2ðtiþkTmÞ�minTm

wL=2ðtiþkTmÞ

!; (26)

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0

10

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500−5

0

5x 10

−3

0 1 2 3 4 5 6 7 8 9 10

10−10

10 −8

10−12

t/T [-]

t/T [-]

u L [-

]w

L/2

[-]

PSD

(w

L/2

)[H

z−1 ]

F/f [-]

Fig. 8. Response for vd=0.26 V, f=f2 and e1=0.5 [–] in terms of wL/2, uL and the power spectral density of wL/2.

170 175 180 185 190 1954

5

6

7

8x 10

−3

170 175 180 185 190 19510

−2

100

102

f [Hz]

Um

[-]

Wm [

-]

up

down

Fig. 9. Frequency–amplitude plot based on frequency sweep analysis for e1=0.5 and vd=0.26 V (simulation).



where Tm = (Ne�Nt)T/Nm and Nm=5 [–]. Note that for harmonic responses, ~Um is equivalent with Um defined by Eq. (23).Time ti indicates the time at which the i th incremental parameter change takes place. Exactly the same procedure with thesame settings for Ne, Nm and Nt will be followed during the experiments.

Results for the numerical frequency sweep for e1=0.5 and vd ¼ 0:264vcd are shown in Fig. 9. Note that ~W m is plotted on

a logarithmic scale. Very severe out-of-plane responses are suddenly initiated at the boundaries of the region182r f r185, where no stable harmonic solutions are found with the continuation approach, see Fig. 7. Note thatfrequency hysteresis is not observed in Fig. 9.

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3

3.5

4x 10

−3

vd = 0.15

vd = 0.14

f [Hz]

A

Um

[-]

stableunstable


2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500

−2

0

2

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500

−2

0

2

x 10−3

0 1 2 3 4 5 6 7 8 9 10

10−10

10−8

10−12

t/T [-]

t/T [-]

u L [

-]w

L/2

[-]

F/f [-]

PSD

(w

L/2

)[H

z−1 ]

Fig. 11. Response for vd = 0.15 V, e3=0.5 and f=183 Hz in terms of wL/2, uL, and the power spectral density of wL/2.



Next, an imperfection of the form e3=0.5 is considered. Using continuation of periodic solutions for this case, asomewhat different scenario is found in the top of the resonance near f= f2, see Fig. 10. First of all, the harmonic responseloses stability at a much lower value of vd, i.e. vc

d ¼ 0:145 V. Furthermore, for vd4vcd two regions of instability appear.

Between these two regions, a small branch with stable harmonic solutions is found, see enlargement A of Fig. 10. Anexample of the response in the right region of instability at f=183 Hz is depicted in Fig. 11. As can be noted, the PSD of thisresponse shows many discrete peaks, which are equally spaced at DF=f ¼ 1=11, suggesting that this beating response is an1/11 subharmonic response. By plotting T-sampled values of uL against T-sampled values of wL/2, a Poincare section of thisresponse is constructed, see the right plot of Fig. 13. The Poincare section shows 11 dots, confirming the 1/11 subharmonicnature of the response. In the left region of instability, a different type of response occurs at f=179 Hz, see Fig. 12. The PSDof this response is broad-banded, suggesting that the response now has a chaotic nature. This suggestion is supported bythe fact that the Poincare map of this response is a bounded cloud of points, see the left plot of Fig. 13. A numericalfrequency sweep analysis for e3=0.5 and vd ¼ 0:154vc

d [V] reveals that in both regions, where no stable harmonic solutionsare obtained, again severe responses appear instead, see the left plots of Fig. 14. As an indication, for the results depicted inFigs. 11 and 12, transversal accelerations in terms of €wðL=2Þ of order 5000g are found. The right plots of Fig. 14 show thatfor a slightly higher value of the input voltage amplitude (vd=0.17 V), the region of instability increases in width and alsostarts to show softening behaviour.

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2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500

−2

0

2

x 10−3

0 1 2 3 4 5 6 7 8 9 10

10−8

10−10

10−12

t/T [-]

t/T [-]

uL [

-]w

L/2

[-]

F/f [-]

PSD

(w

L/2

)[H

z−1 ]

Fig. 12. Response for vd=0.15 V, e3=0.5 and f=179 Hz in terms of wL/2, uL, and the power spectral density of wL/2.

−0.0235 −0.023 −0.0225 −0.022 −0.0215 −0.021−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4f = 179 [Hz]

uL [-] uL [-]

wL

/2 [-

]

−0.022 −0.021 −0.02 −0.019 −0.018 −0.017 −0.016−0.8

−0.6

−0.4

−0.2

0

0.2f = 183 [Hz]

wL

/2 [-

]

Fig. 13. Poincare plots of responses shown in Fig. 12 (left) and in Fig. 11 (right).



The frequency–amplitude plot for an imperfection of the form e5=0.5 will be discussed now, see Fig. 15. For this case,the critical amplitude for the input voltage is even lower, i.e. vd

c =0.135 V. For vd4vcd, a region appears on the left hand side

of f= f2, where two stable harmonic solutions coexist. The frequency sweep analysis results for this case are shown in theleft plots of Fig. 16. Near f=175 and 183 Hz sudden jumps can be seen in the stepped sine results in terms of ~W m. One ofthese jumps occurs in the small frequency region where no stable harmonic solutions exist, see enlargement A in Fig. 15. Inthis small region, beating responses are found comparable to the response as shown in Fig. 12. For a slightly higher value ofvd (vd=0.16 V), again the region of instability increases, while the response in terms of ~W m significantly increases inamplitude, see the right plots of Fig. 16.

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2.5

3

3.5

4

x 10−3

165 170 175 180 185 190 1950

1

2

3

4

165 170 175 180 185 190 1952

2.5

3

3.5

4

x 10−3

165 170 175 180 185 190 1950

1

2

3

4

f [Hz]f [Hz]

vd = 0.15 [V] vd = 0.17 [V]

Um

[-]

Wm

[-]

downup

Fig. 14. Frequency–amplitude plots based on frequency sweep analysis for vd=0.15 V (left) and for vd=0.17 V (right); in both plots e3=0.5.

170 175 180 185 190 1952.2

2.4

2.6

2.8

3

3.2

3.4

3.6

x 10−3

vd = 0.13

vd = 0.14

f [Hz]

Um

[-]

stable

unstable

A




Finally, the influence of the considered circumferential wavenumber n on the critical amplitude of the input voltage (vdc)

is examined. In Fig. 17, the influence of n on the value of vdc is depicted for three imperfection shapes. For the considered

range of n, the lowest value of vdc is obtained for n=9 and e5=0.5, while vd

c is more than a factor two higher for n=5 ande1=0.5. This indicates that, similar to the case of prescribed base-acceleration [5], the obtained critical value highlydepends on the imperfection shape. Furthermore, also in analogy with the results obtained for the case of prescribed base-acceleration, the lowest obtained critical amplitudes of the input voltage are closely grouped together. As stated before,this may suggest that if one would include additional DOFs to the model corresponding to modes with differentcircumferential wavenumbers (or perform an experiment on a real cylindrical shell), multiple modes may start to interactfor vd4vc

d, leading to even more complicated dynamics than observed in this section.

6. Experimental steady-state analysis

In this section, experimental results obtained for the shaker excited cylindrical shell with top mass will be discussed.The experimental steady-state results are obtained for a varying excitation frequency and a varying excitation amplitude

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2.5

3

3.5

4

x 10−3

165 170 175 180 185 190 1950

2

4

6

165 170 175 180 185 190 1951

2

3

4

x 10−3

165 170 175 180 185 190 1950

2

4

6

f [Hz] f [Hz]

vd = 0.14 [V] vd = 0.16 [V]

Um

[-]

Wm

[-]

downup

Fig. 16. Frequency–amplitude plots based on frequency sweep analysis for e5=0.5 and vd=0.14 V (left) and for vd=0.16 V (right).

5 6 7 8 9 10 11 120.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

n [-]

e1 = 0.5

e3 = 0.5

e5 = 0.5

vc d

[V]

Fig. 17. Influence of circumferential wavenumber n and imperfection ei on vdc .



using a stepped sine procedure. During the previously presented numerical analyses, wL/2 was evaluated at the location,where the largest out-of-plane displacements occurred, see Eq. (24). For the experimental analysis it is unknown, wherethe largest out-of-plane displacements will occur. Therefore, the angular location for measuring the out-of-planedisplacement of the shell is chosen by testing initially a number of angular positions along the circumference of thecylindrical shell at height x=L/2, see Fig. 3. The location, where the largest displacements are measured, is used duringthe actual experiments. The measure ~Um is computed using the LVDT measurements, see Fig. 3.

6.1. Results

In Fig. 18, stepped frequency sweep results for various values of vd and Df ¼ 0:5 Hz are depicted. As can be noted, for thesmallest considered amplitude of the input voltage (vd=0.02 V), a single resonance peak appears near f=177 Hz close to

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

120 130 140 150 160 170 180 190 200 210 220 230 2400

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10−3

Um

[-]

Wm

[-]

A

vd = 0.14

vd = 0.08

vd = 0.02

vd = 0.04

vd = 0.06

f [Hz]

Fig. 18. Frequency sweep results (‘+’ sweep down, ‘ 3’ sweep up).



f2=182 Hz. By increasing the value of vd, the amplitude of this resonance peak increases, the resonance frequency decreasessomewhat, and additional peaks appear close to this peak. Time histories of the steady-state response for f=170 Hz interms of wL/2 and their PSDs are depicted in Figs. 19 and 20 for, respectively, vd = 0.06 and 0.08 V. To minimize the effect ofmeasurement noise, each PSD is averaged over eight sets of 2048 data points, measured using a sample frequency of 4 kHzand an anti-aliasing filter. For vd=0.06 V, the PSD is dominated by peaks at integer multiples of the excitation frequency(F/f=1,2,3,y) indicating a harmonic response. However, for the slightly higher value vd=0.08 V, additional frequencycomponents appear in the PSD, which exceed the measurement noise level. The time history shown in Fig. 20 seems to bequasi-periodic or slightly chaotic. The transition from the harmonic response to the aperiodic response occurs without a(noticeable) sudden increase in out-of-plane vibrations, i.e. the frequency–amplitude plot does not exhibit jumps for thisvalue of vd, see Fig. 18.

By increasing the excitation amplitude further to vd=0.14, the frequency–amplitude plot starts to exhibit jumps nearf=162 Hz and near f=170 Hz, see enlargement A in Fig. 18. Especially the jump near f=170 Hz rapidly increases, if vd isincreased even further, see the results for vd=0.2 V in Fig. 21. Furthermore, several small transition regions can be

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−0.2

−0.1

0

0.1

0.2

0 1 2 3 4 5 6 7 8 9 1010

−20

10−10

t/T [-]

F/f [-]

PSD

(w

L/2

)[H

z−1 ]

wL

/2 [

-]

Fig. 19. Measured dimensionless out-of-plane displacement (wL/2, top) and corresponding PSD (bottom) for f=170 Hz and vd=0.06 V.

0 50 100 150 200

−0.2

−0.1

0

0.1

0.2

0 1 2 3 4 5 6 7 8 9 1010−20

10−10

F/f [-]

PSD

(w

L/2

)[H

z−1 ]

wL

/2 [

-]

t/T [-]

Fig. 20. Measured dimensionless out-of-plane displacement wL/2 (top) and corresponding PSD (bottom) for f=170 Hz and vd=0.08 V.



observed, see enlargements A and B in Fig. 21. For this level of vd, the jump near f=170 Hz can also very clearly be notedaudibly. More specifically, for the sweep down, the noise produced by the cylindrical shell suddenly significantly increasesafter passing the jump. By comparing the time histories of the sweep-down response in terms of wL/2 in Fig. 22, i.e. atf=170 Hz just before the large jump, and in Fig. 23, i.e. at f=167.5 Hz just after the large jump, a significant change of theresponse can be noted. At f=167.5 Hz, the response not only has a much larger amplitude than the response at f=170 Hz,but also has a much broader PSD compared to the PSD measured at f=170 Hz.

6.2. Discussion

Experimentally obtained frequency–amplitude results have been presented for increasing values of the excitationamplitude vd. Similar as found in the numerical analyses in Section 5, the experiments show a transition from a harmonic

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0.2

0.4

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0.8

1

1.2

1.4

1.6

120 140 160 180 200 220 2400

1

2

3

4

5

6

7x 10−3

Um

[-]

Wm

[-]

A

B

f [Hz]

Fig. 21. Frequency sweep results for vd=0.2 V (‘+’ sweep down, ‘ 3’ sweep up).



response to an aperiodic large amplitude response near the resonance at f= f2 [Hz]. However, in the experiments forlow values of vd, first additional peaks start to appear near the resonance at f= f2, see Fig. 18. In this frequency regiontransitions are found, where the response switches from a harmonic response to an aperiodic response, without a(noticeable) sudden increase of the out-of-plane vibrations of the shell. Such behaviour is not found in the semi-analyticalanalysis. For larger amplitudes of the harmonic input voltage, the experimental frequency–amplitude plot starts to exhibitjumps near f=162 and 170 Hz for vdZ0:14 V. By increasing vd further to vdZ0:19 V, large amplitude responses with a verybroad PSD appear for excitation frequencies near f=167.5 Hz, see Fig. 23. In the numerical analyses such transitions arefound for values of vd starting near 0.135 V going up to 0.253 V, depending on the shape of the imperfection. Taking intoconsideration that the critical value for vd is very sensitive to imperfections, see Section 5, it can be concluded that thetransitions to the large amplitude, aperiodic responses in the experiments occur at more or less the same values of vd, aspredicted by the semi-analytical approach.

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−1

0

1

2

0 2 4 6 8 10

10

10

−10

−20

t/T [-]

F/f [-]

PSD

(w

L/2

) [H

z−1 ]

wL

/2 [

-]

Fig. 22. Measured dimensionless out-of-plane displacement wL/2 (top) and corresponding PSD (bottom) for f=170 Hz and vd=0.20 V.

0 50 100 150 200 250 300 350 400−2

−1

0

1

2

0 2 4 6 8 10

10

10

−10

−20

t/T [-]

F/f [-]

PSD

(w

L/2

)[H

z−1 ]

wL

/2 [-

]

Fig. 23. Measured dimensionless out-of-plane displacement wL/2 (top) and corresponding PSD (bottom) for f=167.5 Hz and vd=0.20 V.



There may be various reasons for the remaining (quantitative) mismatches between the semi-analytical results and theexperimental results. Firstly, the mismatch may be explained by the imperfection sensitivity of the post-critical responseas illustrated in the semi-analytical analysis, compare for example Figs. 9, 14, and 16. The cylindrical shell used in theexperiments will obviously not be geometrically perfect and its actual (unknown) radial imperfection shape will bedifferent from the imperfection shapes considered in the semi-analytical analysis. Furthermore, at the experimental setup,also other types of imperfections will be present, for example small misalignments between the lower and the uppersledge, see Fig. 3, small thickness variations of the shell and non-perfect shell clamping conditions. As a second cause,arbitrary imperfections in the shell and/or in its boundary conditions will simultaneously trigger multiple axi-asymmetrical modes with different circumferential wavenumbers n. The effect of the participation of multiple modes withdifferent circumferential wavenumbers is not examined in the semi-analytical analysis. However, the effect of variation inthe circumferential wavenumber n on the dynamic stability has been considered in the current analysis for severalimperfections, see Fig. 17. From these analyses it follows that the lowest obtained critical amplitudes of the input voltageare closely grouped together for different values of n. This suggests that in case of arbitrary imperfections, multiple modeswith different circumferential wavenumbers may start to interact for vd4vc

d. To examine such responses using the semi-analytical approach, more DOFs, corresponding to modes with other circumferential wavenumbers, should be included inthe expansion of w and w0, see Eqs. (12) and (13).

7. Conclusions

The dynamic stability limits of a thin cylindrical orthotropic shell with top mass, which is axially excited at its base byan electrodynamic shaker, are determined both numerically and experimentally. To be able to compare the experimentalresults with the semi-analytical results, a coupled shaker-structure model is derived. The model is validated firstly bycomparing numerical and experimental results for the FRF representing the top mass acceleration response caused by theinput voltage, and secondly by comparing experimental eigenfrequencies with eigenfrequencies of linearized models.

The coupled structure exhibits two resonances, which occur at frequencies, which are low compared to resonancefrequencies of the cylindrical shell. The first resonance frequency at f1 [Hz] corresponds to a suspension mode of the shakermass. The second resonance frequency at f2 [Hz] corresponds to a suspension mode, which is dominated by axial vibrationsof the (cylindrical shell with) top mass. Numerical analyses show that the harmonic response may become unstable nearthe second resonance frequency and an aperiodic response with severe out-of-plane deformations may appear instead. Thecritical value for the input voltage, for which the harmonic response changes to the severe post-critical response, highlydepends on the initial imperfections present in the shell.

Experiments qualitatively confirm the dynamic response predicted by the semi-analytical model (including the shakerdynamics). Indeed, in the experiments it is shown that, by increasing the excitation amplitude of the harmonic inputvoltage near the frequency f2 [Hz], the harmonic response may switch to a severe aperiodic response. A quantitative matchbetween the experimental and the semi-analytical results is not realized yet. In order to achieve this, it is important toextend the semi-analytical model with modes corresponding to multiple circumferential wavenumbers. Moreover, it willbe necessary to measure the actual imperfection of the shell under experimental investigation, so that this imperfectioncan be included in the semi-analytical model.

The presented experimental results confirm the observation from the semi-analytical analysis, that the dynamicstability analysis of base-excited cylindrical shells structures with top mass should be concentrated near a resonance ofrelatively low frequency, corresponding to an axi-symmetrical suspension mode, dominated by axial vibrations of the(cylindrical shell with) top mass.

References

[1] P. Gonc-alves, Z.D. Prado, Nonlinear oscillations and stability of parametrically excited cylindrical shells, Meccanica 37 (6) (2002) 569–597.[2] M. Amabili, M. Paıdoussis, Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and

without fluid–structure interaction, Appl. Mech. Rev. 56 (4) (2003) 349–356.[3] E. Jansen, Non-stationary flexural vibration behaviour of a cylindrical shell, Int. J. Non-Linear Mech. 37 (4–5) (2002) 937–949.[4] F. Pellicano, M. Amabili, Stability and vibration of empty and fluid-filled circular shells under static and periodic axial loads, Int. J. Solids Struct. 40

(13–14) (2003) 3229–3251.[5] N. Mallon, R. Fey, H. Nijmeijer, Dynamic stability of a thin cylindrical shell with top mass subjected to harmonic base-acceleration, Int. J. Solids Struct.

45 (6) (2008) 1587–1613.[6] F. Pellicano, Experimental analysis of seismically excited circular cylindrical shells, Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference,

Eindhoven, the Netherlands, 2005.[7] S. Zhang, J. Li, Anisotropic elastic moduli and Poisson’s ratios of a poly(ethylene terephthalate) film, J. Polym. Sci. Part B Polym. Phys. 42 (3) (2003)

260–266.[8] N. Mallon, Dynamic Stability of Thin-walled Structures: A Semi-analytical and Experimental Approach, PhD Thesis, Eindhoven University of

Technology, 2008.[9] T. Koga, Effects of boundary conditions on the free vibrations of circular cylindrical shells, AIAA J. 26 (11) (1988) 1387–1394.

[10] D. Liu, Nonlinear Vibrations of Imperfect Thin-walled Cylindrical Shells, PhD Thesis, Delft University of Technology, 1988.[11] M. Amabili, Nonlinear vibrations of circular cylindrical shells with different boundary conditions, AIAA J. 41 (6) (2003) 1119–1130.[12] /http://www.tuedacs.nl/S, TUeDACS Advanced Quadrature Interface.[13] D. Brush, B. Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York, 1975.[14] L. Donnell, Beams, Plates and Shells, McGraw-Hill, New York, 1976.

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[15] D.S. Lee, Nonlinear dynamic buckling of orthotropic cylindrical shells subjected to rapidly applied loads, J. Eng. Math. 38 (2000) 141–154.[16] X. Li, Y. Chen, Transient dynamic response analysis of orthotropic circular cylindrical shell under external hydrostatic pressure, J. Sound Vibr. 257 (5)

(2002) 967–976.[17] T. Ng, K. Lam, Effects of boundary conditions on the parametric resonance of cylindrical shells under axial loading, Shock Vibr. 5 (1998) 343–354.[18] M. Amabili, Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections, J. Sound

Vibr. 262 (4) (2003) 921–975.[19] X. Xu, E. Pavlovskaia, M. Wiercigroch, F. Romeo, S. Lenci, Dynamic interactions between parametric pendulum and electro-dynamical shaker,

Z. Angew. Math. Mech. 87 (2) (2007) 172–186.[20] K. McConnell, Vibration Testing, Theory and Practice, Wiley, New York, 1995.[21] A. Preumont, Mechatronics, Dynamics of Electromechanical and Piezoelectric Systems, Springer, Berlin, 2006.[22] MSC.Software corporation, MSC.Marc manual Volume B, Element Library, 2005.[23] L. den Boer, Modal analysis of a thin cylindrical shell with top mass, Technical Report DCT 2007.086, Eindhoven University of Technology, 2007.[24] R.D. Zou, C.G. Foster, Simple solution for buckling of orthotropic circular cylindrical shells, Thin-Walled Struct. 22 (3) (1995) 143–158.[25] N. Yamaki, Elastic Stability of Circular Cylindrical Shells, Elsevier Science Publishers, Amsterdam, 1987.[26] E. Doedel, R. Paffenroth, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, B. Sandstede, X. Wang, AUTO97: Continuation and bifurcation

software for ordinary differential equations (with HOMCONT), Technical Report, Concordia University, 1998.

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